To deconstruct the genius of Bach, to fathom how the cold math of line plotted against line, note riding against note, voices knitted into voices, can translate into sounds often held up as the very pinnacle of Western music, to explain the whole history of a composer who the history books insist “invented” musical grammar but whose reputation evaporated from view for a hundred years after his death in 1750 – the name “Bach” meaning a famous teacher and organist to most people living in the early 1800s – to view Bach not through the prism of our twenty-first century minds, where we might mistakenly assume that the lifestyle, function and expectations of a composer were the same as today, but to place Bach in the right historical context, could take some kind of genius in itself.

Or perhaps not. Wrapped up in the mystery of Johann Sebastian Bach is his very familiarity. Once you’ve internalized the lessons of harmony and counterpoint that Bach formalized in the near-200 chorale harmonizations he wrote throughout his life and in works like the The Well-Tempered Clavier (BWV 846-93) – the so-called forty-eight; two books each made up from a prelude and fugue in all twenty-four major and minor keys – practically every note he composed can be slotted neatly into his rational and consistent system. Familiarity is bred from an early age. Every night my two- year old son goes to bed, his music-box offers two choices: sounds of nature or Bach, the inference being that at some deep human level they have become interchangeable. And if, one day, my son goes to music college, those same Bachian principles of harmony and counterpoint will be hardwired into his consciousness like, at primary school, the alphabet, or the reliable simplicity that one plus one is always going to equal two.

Theoretically interpreting and making sense of Bach ought to be as straightforward and user-friendly as assembling an IKEA bookcase: begin with the component parts, follow the manual, and you can’t go far wrong. And a door opens on perhaps Bach’s most profound enigma. Musicians can actively hear the harmonic processes of Bach clearly and unambiguously functioning in front of their ears – unlike Haydn, Beethoven or Bruckner there are no blots from the blue. These harmonic patterns are deeply woven inside our cultural DNA. Where would the Scherzo from Mahler’s Resurrection Symphony, Stravinsky’s Dumbarton Oaks, The Kinks’ Village Green, the forward-thinking jazz of Dave Brubeck and Lennie Tristano, or The Beach Boys’ Lady Lynda have been without Bach? And yet it’s entirely possible to play all the notes devotedly and still get the music wrong. There’s a part of Bach we can’t have. One plus one might always make two, but Bach’s music is interested in the mysteries of why.

The leading British conductor and Bach scholar Christopher Hogwood, who in 1973 founded the Academy of Ancient Music with its mission to play Baroque music on period instruments, tells me that he’s puzzled by students coming his way who, for instance, play minuets every day of their lives without knowing how to dance a minuet. “That doesn’t mean they don’t play a charming minuet,” he says, “but trying to make sense of Bach without knowing what was in his world is a compromise. I understand that students who practice their instruments for eight hours a day are unlikely to want to go to the library to learn about eighteenth century theology. But there’s no point in playing a chorale prelude without knowing the chorale. And if you know the chorale you might as well know the words that were sung to the chorale; and then you might as well know a little bit about eighteenth-century theology, Lutherism and Calvinism, and you’ll be a little closer to what was in Bach’s world.”

And Hogwood is keen to press another distinction about the distance between then and now which knocks back on the sort of compositional material Bach generated and worked with. Interpreters take note. “All music then was contemporary music,” he explains. “You wrote to be played tomorrow and you forgot about it the day after. It was very immediate and if there was no performance, or the opportunity suddenly collapsed, you simply stopped writing. People didn’t want to hear something that was a year old, certainly not ten years old, and never a century old. Composers were workers, employed on the same terms as the cook, or the coachman, or the gardener. You didn’t always require to know the name of the gardener, but if you became a well-known gardener people might come to look at your garden in the same way people came to Venice to hear Vivaldi. But very few people came to hear Bach. He never got a top job and was isolated – and knew it.”

Hogwood talks about the pressure on Bach to crank out a fresh cantata every Sunday. And with his wife and sons lined up to copy parts and fill out Bach’s harmonies – applying those forever internally consistent harmonic procedures – the sheer industry of his art becomes clear. The bottom drawer was regularly and unapologetically plundered. Up against an impossible deadline? The Brandenburg Concerto no. 3 in G Major(BWV 1048), with added chorus, becomes that Sunday’s cantata. (“You don’t have any sense that a chorus is ‘missing’,” Hogwood muses, “but Bach certainly had a sense that one could be added.”) Practicality, recycling, the brutal craft of needing to have his cantata ready each Sunday was everything.

Which means Bach needed his material to be bulletproof; self-generative processes, like canons and fugues, once triggered, had to slot together and move forward with the architectural logic of a subway map. No time for unpicking, correcting or finessing. Bach was a servant writing music for the greater glory of God. Move forwards a century and Beethoven’s Missa solemnis is a dialogue with the divine, albeit an essentially God-fearing one. Beethoven’s great works – the Symphony no. 5 in C minor, the Violin Concerto in D Major, his Opus 111 Piano Sonata in C minor – are dialogues with a world that has Beethoven and his obsessions at its center. The techniques of harmony and counterpoint he inherited from Bach are re-sculpted, re-constituted, thought through afresh. Each piece requires a new solution, part musical and part philosophical, that could not be turned around on weekly cycle. Which doesn’t mean Beethoven couldn’t have worked under pressure. But he opted not to – patronage had switched from the church to wealthy individuals and secular organizations. Beethoven was no servant; he was an “artist” in a sense Bach would not have understood.

The modern construct supposes that Bach himself was divine, which on some level may or may not be true, but it’s not an idea that would have pleased him. His work was an attempt to deal with, give voice to, offer some humble explanation for, worlds beyond this one. The personalities and experiences of Beethoven and Mahler understandably became part of the story: the frustrations of a deaf composer, the terror of heart disease makes good copy. But Bach as physical, living presence was unimportant to the notes he put on the page. A cool, emotionally objectifying distance exists between Bach and his material; beauty and emotional resonance, rather like in the music of Varèse or Xenakis, is found in the high-intelligent design of structure, proportion and inner-order.

Bach’s Goldberg Variations (BWV 988), a sequence of thirty variations on the bass line of an aria, written near the end of his life in 1741, has been endlessly analyzed, line by line, note by note, voice by voice, as music object, mathematical phenomenon and cultural icon. The work that haunted the eternally haunted Glenn Gould and bookended his recording career – cue Romanticized, rock-star idolatry – has also been reversed-engineered by musicologists with the plucky determination of scientists trying to whistleblow the formula for Coca-Cola. Bach’s proportional arithmetic, apparently, proves an irresistible draw. Certain features reoccur, structural markers in time. Every third variation is a canon, and each canon progressively imitates at a step further along the scale. The surrounding variations alternate between generic forms – dances, arias, a fughetta and at the mid-point a stately French Overture – and quick, freer form variations. How deeply performers need to grasp these underpinning numerical relationships is an ongoing point of discussion.

Christopher Hogwood is surprisingly phlegmatic. “What’s helpful to a student composer might not be helpful to a player,” he counters as I express some half-baked opinion that he must devote lots of time to counting bars. “You can’t play proportionally, you play what’s in front of you. In music, some mathematical things fall out by default – the Goldberg theme is in a regular number of bars, every variation is the same number of bars, and a mathematical matrix is imposed. More complicated relationships I suspect, yes, were artificial constructs. A number system is a tremendous aid to composers who don’t want to spoil the form of something; artists and architects rely on golden means and Fibonacci series calculations, and composers are no different . . . apart from in one way. Pure proportion with nothing else would be a dull piece of music. You don’t “see” a fugue in one moment, like a painting or a building; music is temporal. It’s pleasing to realize something so well proportioned that it is aesthetically a work of art. But if a piece were to overshoot the Fibonacci series by one bar I’m not certain that would worry most people.”

The jazz pianist, free improviser, composer and onetime classical organist, Oxford-based Alexander Hawkins – who earlier this year premiered a major Bach-inspired commission for jazz musicians on BBC Radio 3, One Tree Found – is clearly more entranced, perhaps even slightly spooked, by the symbolism of Bach’s numerology than Hogwood. As we sit down with the score of the Goldberg Variations, Hawkins turns human calculator. “I’ve always liked,” he reflects, “that the second book begins with a French Overture. It’s nicely perverse having an overture in the middle. And it subtly breaks the regularity of Bach’s maths. This is piece that isn’t sixty-four, or thirty-two, bars long. How long is it? With the repeats it comes out at ninety-five bars – nine plus five equals fourteen; BACH – B is two, A is one, C is three, H is eight, add those numbers together and it comes to fourteen. Bach has embedded his own musical signature into the middle of the mathematical architecture, surely no coincidence.”

By extension, Hawkins tells me, the number five (one plus four) always has significance in Bach, while the number three invariably symbolizes the holy trinity. But Hawkins and Hogwood are in agreement about a wider point: these numerical markers are buried way too deep for performers to communicate their specifics to audiences. “As a performer,” Hawkins says, “you treat the Goldberg Variations with care because you admire the craft and realize things happens for a reason. The maths works on so many levels, but at the same time, the piece wears the arithmetic very lightly. You never listen with the mathematics at the forefront of your mind.” Hogwood draws an analogy with Schoenberg’s serialism. “If it helps a performer to trace the tone rows through a piece of Schoenberg, or reach an understanding of how the maths operates in the Goldbergs then, fine, analyze away. But those relationships will not be audible, and your audience is only interested in what is audible.”

Hawkins’ One Tree Found makes you take notice, quenches your thirsty ears, via its thoughtful riffing off Bach’s palette of techniques and its refusal to go for the easy option – hello Jacques Loussier – of aping Bach’s style. Here’s a performer who has arrived at an understanding of how Bach operated by filtering his fingerprint techniques through other preoccupations. The first section of Hawkins’ piece revisits the idea of canons, but working with improvising musicians required a shift of focus.

“I’m interested in giving musicians leeway,” he elucidates. “There would have been no point in writing a canonic piece – and telling everyone in the program note, hey, my piece is about Pi – if no one could hear Pi. And I asked myself what exactly is the essential idea of a canon? The first time I felt a sense of wonder about canons was in my teens when I played the Canonic Variations on Vom Himmel hoch da komm ich her (BWV 769) and realized, despite everything I’d been taught about parallel and consecutive fourths and fifths being an absolute no-no, here was Bach – Bach! – writing canons at the fourth and fifth and it sounded beautiful. The essential feature of a canon is that material occurs consecutively, out of phase. And in my piece the musicians can move through the material I give them as they wish, improvising their entries. The basic melodic modules are arranged additively (1; 1+2; 2+3+4; 3+4+5+6) and effectively you hear canons both vertically and horizontally, because your ear never quite knows where you are in the process.”

Gottfried Reiche, Bach’s trumpeter

Hawkins projects Bach into the future as a creative going concern; Hogwood tries to strip away layers of accumulated misunderstandings and outmoded ways-of-doing to reach an historically-informed view of how Bach can be played most authentically today, while a musician like the natural trumpet specialist Jonathan Freeman-Attwood has toiled at the coalfield of hard, exploratory, instrumental trial-and-error. Top of the agenda when I meet Freeman-Attwood is Bach’s Brandenburg Concerto no. 2 in F Major (BWV 1047) with its fleet, chromatically devil-may-dare, trumpet writing designed with a clarino trumpeter in mind – a trumpeter who found, lipped and tongued notes without the safety net of valves or holes.

Valve technology would not evolve for another century and Freeman-Attwood continues to be pulled towards what he terms “the raw Pythagorean science” of making music through what amounts to a four-foot length of metal. “The perpetual conflict between pragmatism and idealism is a composer’s lot,” he says, “and we know that Bach regularly wrote music that was too difficult for the forces he had. At times he must have said this makes absolute sense compositionally; I am going to take this fugue to this place, knowing full well that a couple of top trebles aren’t going to be around next Sunday.”

There’s more than a suggestion, Freeman-Attwood says, that the Brandenburg Concerto no. 2 concertante group might originally have consisted of violin, recorder, oboe – and horn rather than trumpet. “The trumpet part was so high, much higher than anything else he’d written for the instrument. Bach never wrote a trumpet part in F [the pitch of the horn] in any other context. It could well have been played an octave lower during Bach’s time. Having said that, some of the concertante dialogues don’t make as much sense without the crystalline spacing of the solo quartet with the trumpet in the stratosphere.”

And the spur to write – or recycle – the Brandenburg Concerto no 2 for trumpet must have coincided with Bach encountering a top-notch clarino virtuoso? “That’s difficult to say. Trumpeters usually had some sort of municipal role, playing fanfares in court and the like, and the good ones were selected to play concert music. What is key, and actually creates the difference in sound in the second Brandenburg Concerto between the modern and old trumpet, comes down to the mouthpiece they used – a considerably larger mouthpiece than today. Their approach to articulation and strength must have been formidable because, today, if we feel a little insecure about high notes we put in a mouthpiece that is slightly shallower, which means you can hit the high notes a little bit easier. In Bach’s day trumpeters must have had something in their diet, or perhaps a special technique, because they played high notes with these huge mouthpieces. We don’t know who Bach had in mind for the second Brandenburg Concerto; but he must have had considerable chops.”

Then we dive into the score, Freeman-Attwood pointing to notes that natural trumpeters would have needed to lip down, plucking notes out of the chromatic ether. The effect, he says, of hearing a natural trumpet play the second Brandenburg Concerto rather than a piccolo trumpet – the modern day alternative – is that you hear a “clucking” rather than a “symphonic” attack. The sound is more coppery than brassy. “Bach is so ingenious that all the notes he uses are in the harmonic series. And here – look! He even dares to go into a minor key. There’s one other piece, by Biber, that has a natural trumpet play in a minor key.”

As a writer whose usual terrain is New Music and jazz, I feel strangely at home discussing a composer who pursues instruments to the very limits of their capability. As we’re wrapping up, Freeman-Attwood discusses the insolvable balance problems that inevitably exist between trumpet, oboe, violin and recorder; Christopher Hogwood goes even further. “It contains some grand music but it’s a failure; I defy you to hear the recorder part when the other three instruments are playing. It looks good on paper but, short of close miking every instrument and falsifying the balance, it’s impossible to bring off in a concert hall.”

And now that we know the world – from macrophage blood cells, to our genetic code, to fractal geometry – is constructed from systems evenly balanced between the rational and chaotic, the science and the acoustics and the intelligent design of Bach has become part of a wider argument. Published in 1979, Gödel, Escher, Bach: An Eternal Golden Braid by the American mathematician and computer scientist Douglas R. Hofstadter refracted Bach’s techniques through the maths of Kurt Gödel and the optical illusion art of MC Escher. “Every aspect of thinking,” Hofstadter writes, “can be viewed as a high-level description of a system which, on a low level is governed by simple, even formal, rules. . . . The image is that of a formal system underlying an ‘informal system’ – a system which can make puns, discover number patterns, forget names, make awful blunders in chess and so forth.” Meanwhile, another scientist, Albert Einstein, left the world in doubt about where he stood in regards to Bach.“I feel uncomfortable listening to Beethoven. I think he is too personal, almost naked. Give me Bach, rather, and then more Bach.”

In the final concert of the thirty-second season of the Boulder Bach Festival, Zachary Carrettin, violin, and Rick Erickson, harpsichord and organ, will join forces as Duo Crezdi in an artist recital at First Congregational Church in Boulder, Colorado at 7:30 pm on Friday, 3 May 2013.

The violin sonatas on the program will come from very different places and times. The earliest is Dario Castello’s Sonata Prima in A minor from Libro secundo. Diligent scholarship has not been able to determine any exact dates for Castello’s birth or death. In fact, almost no shreds of biographical evidence exist about the composer. This much is known: Castello lived much of his life in and around Venice. He was an excellent wind player and a master of the bassoon, an instrument which was very popular in Venice at the time. He published two books of sonatas in 1621 and 1629, which were so popular that they were reprinted in the 1640s and 50s. His sonatas are made up of a number of short contrasting sections and the work to be performed by the Duo Crezdi is no exception. As one of the first to write idiomatically for the violin, Castello’s music has a refreshing and delightful spontaneity about it which is easy to hear even four hundred years later.

Biber’s (1644-1704) Passacaglia stands as the last piece in the composer’s ambitious cycle of Rosary Sonatas. For each of the fifteen Sacred Mysteries, Biber composed a violin sonata, but each one is in a different tuning, or scordatura. In the sole surviving copy of the score, every sonata is prefaced by a lovely copper engraving. The Passacaglia is the last piece in the cycle, and the only one for violin without continuo. It is the only sonata that “duplicates” an earlier tuning for the violin – in this case, the ordinary tuning G – D – A – E. The Passacaglia consists of sixty-five repetitions of a descending tetrachord (four-note motive) in which all manner of harmony, melody and expression appears. In the middle of the piece, the descending motive even appears in an upper voice, complicating matters for the interpreter. Claiming the prize as probably the most elaborate composition for solo violin up to that time, Biber’s Passacaglia almost surely had some influence on J. S. Bach when he composed, some decades later, his own magnificent Chaconne in D minor (BWV 1004) for solo violin.

Chronologically speaking, the Bach and Veracini sonatas date from almost the same time. Bach’s six accompanied sonatas for violin and keyboard were composed no later than 1725, and the composer continued to tinker with them for many years thereafter. The Sonata in C minor (BWV 1017) is cast very much in the mold of a trio sonata. As Peter Wollny noted, each part (violin, keyboard right hand, keyboard left hand) has its own rhythmic character in the slow movements. For example, in the third movement the violin has a lyrical melody, the right hand of the keyboard has continuous triplets, and the left hand has a bass line mostly in quarter notes. The two fast movements feature dense counterpoint and imitative textures, perfectly in keeping with the idea that each part should be interesting and meaningful in and of itself.

Antonio Veracini (1659-1733) led a colorful life and held important posts in Florence and Dresden. As the story goes, he once claimed that “as there is one God, there is one Veracini.” Without lacking a sense of drama either in life or in music, Veracini once jumped out of a building during an argument in Dresden. He even survived a shipwreck in the English Channel. Veracini’s nephew diplomatically wrote that “the heart, rather than cleverness, guided [Veracini’s] finger and bow.” Of his many published violin works, Veracini’s Sonata in G minor (appearing in 1721 as op. 1, no. 1) is an intense, many-colored piece. Opening with a broad French-style introduction, it quickly moves through an impetuous Allegro before settling into a more lyrical Aria. The anxious Allegro that follows contains several outbursts that might have given even the stoic Bach a severe case of indigestion. The final two movements, a short Minuet and an almost ridiculous Gigue that alludes to the sound of the postman’s bell, do little to dispel the image of an undeniably brilliant, yet slightly unstable, musical mind.

Two works for organ on the concert will feature compositions by men who were close to the Bach circle personally. Johann Ludwig Krebs (1713-1780) studied with Bach in Leipzig. The use of the letters of Bach’s name in the fugue subject reflect an ancient practice, dating back to Josquin, called soggetto cavato (“carved subject”) where the letters of a name determine the musical notes. We therefore have, according to German spelling, a subject of B-flat, A, C, B-natural.

Georg Böhm (1661-1733) was connected to the Bach family from his studies in Ohrdruf, a town which knew several generations of Bachs. Böhm might have tutored the young J. S. Bach but there is no direct evidence to support this assertion. Much later, C. P. E. Bach claimed that his father, J. S., loved to study Böhm’s music. Böhm’s Vater unser im Himmelreich is stylistically much like Buxtehude’s music. It is probably one of the most expressive works he penned. Particularly noteworthy is Böhm’s exploration of the organ’s high coloratura register.

The remaining works by J. S. Bach will show his mastery both of small-scale and large-scale musical forms. The Duetto no. 3 in G Major (BWV 804) is similar in texture to one of Bach’s famous Fifteen Inventions (BWV 772-86), but it is more extended and elaborate. Bach derives an entire motivic menagerie by exploring the possibilities of a simple seven-note cell, heard at the beginning in the right hand. Contrasting with this small scale form, the chorale partita for organ on Sei gegrüsset, Jesu gütig (BWV 768) is an extended set of variations on a chorale tune. This is the most ornate of the four sets of chorale variations Bach left us and contains eleven variations on the tune. The variations range from simple to elaborate and give us a full spectrum of Bach’s powers of inventiveness.